The Internet Gave Us Knowledge. Can AI Gives Us Clarity?
When I was in college, I was assigned The Economic and Philosophic Manuscripts by Karl Marx. Unlike The Communist Manifesto, which Engels so politely fixed for us, this book makes little to no sense at first glance.
I was assigned it several times. The first time, I read it at least five times. That’s how long it took before I had even a vague sense of what it was about or could form coherent thoughts.
It might be that I’m an idiot — but I don’t think that was the case.
The one time I was assigned Discipline and Punish by Foucault, I read it three times and still never figured out what he was on about.
As someone with a degree in philosophy, I have maybe a hundred examples of reading a book over and over, or frantically writing down everything a professor said, trying to decipher it later.
I still take notes that way. If you saw my notes after an interview with you, you'd see I wrote down almost every third word you said. For example:
"People that clicked, filled the form."
This isn’t useful context about a person’s talent. It’s just a literal description of what the UI did. I didn’t grade them on it — but I wrote it down anyway.
What I don’t do anymore is sit there, stuck, trying to guess what something means.
When I read Dark Pools and felt lost because the book felt floaty, I didn’t start over. I asked a chatbot to explain the part I didn’t get.
Earlier today, I was watching an academic video on probability analysis and typed:
bayes theorem plz
That option didn’t exist 10 or 15 years ago. I couldn’t try, get it wrong, and ask again with a clearer question.
When I was 15, I went to my Algebra II teacher and asked her why there wasn’t a function for the n-th Fibonacci number. And bless her — she did her best.
But here’s what I would’ve gotten today:
> why isn't there a function for the nth digit in the Fibonacci sequence that isn't recursive?
Great question — and there **is** a way to compute the *n-th Fibonacci number* directly **without recursion**, using **closed-form expressions** or **iterative algorithms**.
### 1. **Closed-form (Binet’s Formula)** – no recursion
There's a math formula for the *n-th Fibonacci number*:
$$
F(n) = \frac{1}{\sqrt{5}} \left( \phi^n - \psi^n \right)
$$
Where:
* $\phi = \frac{1 + \sqrt{5}}{2}$ (the golden ratio)
* $\psi = \frac{1 - \sqrt{5}}{2}$
This is called **Binet’s formula**. It’s **not recursive**, but it uses floating-point math, so it’s inaccurate for large $n$ due to rounding.
---
### 2. **Iterative algorithm** – also not recursive
python
def fib(n):
a, b = 0, 1
for _ in range(n):
a, b = b, a + b
return a
This is **efficient**, accurate, and also not recursive.
---
### 3. **Matrix exponentiation** – fast and non-recursive
Fibonacci numbers grow like powers of a matrix:
$$
\begin{bmatrix}
1 & 1 \\
1 & 0
\end{bmatrix}^n =
\begin{bmatrix}
F(n+1) & F(n) \\
F(n) & F(n-1)
\end{bmatrix}
$$
Raising this matrix to the \$n\$-th power using exponentiation by squaring gives you the Fibonacci number in \$O(\log n)\$ time — also not recursive.
---
So in short: **recursive Fibonacci is simple but inefficient**, and there are **non-recursive alternatives** that are widely used in real applications.
And I think this changes everything — not in the "everyone is plagiarizing their homework" way.
The internet made knowledge accessible. AI might make understanding possible.